Current conservation as a geometric property of space–time

Abstract

Working within the framework of stochastic quantum mechanics, we prove that a mild reality constraint on the radial part of the resolution generator renders the (probability) current, for a single particle with only positive (or only negative) energies and having arbitrary integer spin, conserved. The result is proven both nonrelativistically and relativistically. This extends some earlier results that had been obtained for spinless systems. We argue that the origin of such a conservation law is a consequence of the particular geometry of space–time predicated by the formalism of stochastic quantum mechanics.